Material for : Dynamic dissipative cooling of a mechanical oscillator in strong-coupling optomechanics

This Supplementary Material is organized as follows. In Sec. I, we present the linearization of the optomechanical Hamiltonian. Using the quantum master equation, we describe the covariance approach to calculate the mean phonon number in Sec. II. With this approach, we derive the steady-state cooling limit in Sec. III and the time evolution of mean phonon number in Sec. IV. In Sec. V, we calculate the instantaneous-state cooling limit in the strong coupling regime and optimize the result according to the frequency matching condition. In Sec. VI, the dynamic modulation of cavity dissipation is introduced. In Sec. VII, we discuss the dissipation quantum backaction and interaction quantum backaction. In Sec. VIII we investigate the effect of dissipation pulsewidth. The discussion on the bad-cavity limit is studied in Sec. IX.